Suppose f is differentiable on (-∞,∞), f(1) = 2, and f'(1) = 3...

Question

Suppose f is differentiable on (-∞,∞), f(1) = 2, and f'(1) = 3. Find the linear approximation to f at x = 1 and use it to approximate f (1.1).

Accepted Answer

Hi everyone, let's take a look at this problem dealing with linear approximation. This problem says consider a differentiable function G on the open interval from minus affinity to infinity, where G of 2 is equal to 3 and G 2 is equal to minus 2. Use the linear approximation to estimate the value of G of 2.05. We in four possible choices as our answers. For choice A, we have 1.9. For choice B, we have 2.9. for choice C, we have 3.9, and for choice D, we have 3.1. Now, we're asked to find the value of G of 2.05 using a linear approximation. So we, we call for our linear approximation that we will have L of X. is equal to GFA. Plus G A multiplied by the quantity of X minus A. Now, in our problem here, A is going to be equal to 2 since we were given the values of G 2 as well as G2, and we're looking for the value of G of 2.05, so that means that X is going to be equal to 2.05. So we'll substitute in those values into our expression, and so we will have L of 2.05. is equal to G2, which we were given in the problem, has a value of 3, then we'll have plus. G 2, which has a value of minus 2, so we'll have -2 multiplied by the quantity of X minus A, which is 2.05 minus 2. So simplifying this expression, this is going to be equal to 3 minus 0.1, which is equal to 2.9. So this here is the answer that we're looking for for this problem, and if we compare that to our answer choices, we see that the correct answer choice corresponds to answer choice B. So I hope that this explanation has been useful, and I'll see you in the next video.

Suppose f is differentiable on (-∞,∞), f(1) = 2, and f'(1) = 3. Find the linear approximation to f at x = 1 and use it to approximate f (1.1).

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